Modeling the Footprint and Equivalent Radiance Transfer Path Length for Tower-Based Hemispherical Observations of Chlorophyll Fluorescence

The measurement of solar-induced chlorophyll fluorescence (SIF) is a new tool for estimating gross primary production (GPP). Continuous tower-based spectral observations together with flux measurements are an efficient way of linking the SIF to the GPP. Compared to conical observations, hemispherical observations made with cosine-corrected foreoptic have a much larger field of view and can better match the footprint of the tower-based flux measurements. However, estimating the equivalent radiation transfer path length (ERTPL) for hemispherical observations is more complex than for conical observations and this is a key problem that needs to be addressed before accurate retrieval of SIF can be made. In this paper, we first modeled the footprint of hemispherical spectral measurements and found that, under convective conditions with light winds, 90% of the total radiation came from an FOV of width 72°, which in turn covered 75.68% of the source area of the flux measurements. In contrast, conical spectral observations covered only 1.93% of the flux footprint. Secondly, using theoretical considerations, we modeled the ERTPL of the hemispherical spectral observations made with cosine-corrected foreoptic and found that the ERTPL was approximately equal to twice the sensor height above the canopy. Finally, the modeled ERTPL was evaluated using a simulated dataset. The ERTPL calculated using the simulated data was about 1.89 times the sensor’s height above the target surface, which was quite close to the results for the modeled ERTPL. Furthermore, the SIF retrieved from atmospherically corrected spectra using the modeled ERTPL fitted well with the reference values, giving a relative root mean square error of 18.22%. These results show that the modeled ERTPL was reasonable and that this method is applicable to tower-based hemispherical observations of SIF.


Introduction
Accurate estimation of gross primary production (GPP) is of great importance to global change research and ecosystem monitoring. In recent years, the solar-induced chlorophyll fluorescence (SIF) retrieved from hyperspectral data has become a new powerful tool for the estimation of the ecosystem GPP [1][2][3][4][5][6].
The eddy covariance (EC) technique has dramatically enhanced our understanding of interand intra-annual variations in carbon fluxes at the ecosystem scale [7]. However, EC measurements are dispersed and only cover quite limited regions [8]. For continuous global monitoring of carbon to the study by Liu et al. [27], a bias of 10 m in the sensor's height will lead to a bias of about 0.1 mW/m 2 /nm/sr in the retrieved SIF at the O 2 -A band, which is not negligible for SIF retrieval. Therefore, accurate atmospheric correction (in other words, accurate estimation of the ERTPL) is of vital importance for tower-based observations of SIF, even if the height of the sensor above the canopy is only some tens of meters. Consequently, the ERTPL of bi-hemispherical measurements must be analyzed. However, according to our knowledge, no such analysis has been reported until now.
Sensors 2017, 17,1131 3 of 15 canopy is only some tens of meters. Consequently, the ERTPL of bi-hemispherical measurements must be analyzed. However, according to our knowledge, no such analysis has been reported until now.
Bare fiber foreoptic Spectrometer Cosine-corrected foreoptic Therefore, the modeling of the footprint and ERTPL of bi-hemispherical spectral measurements is important in the context of tower-based measurements of SIF. The aims of this paper are: (1) to analyze the footprints of tower-based hemispherical spectral observations, and also the match between these footprints and those of tower-based flux observations; (2) to determine the ERTPL of hemispherical spectral observations and; (3) to evaluate the derived ERTPL using simulations and to test the validity of the atmospheric correction by using the derived ERTPL for tower-based SIF retrieval.

Data Simulation
In order to model the ERTPL, we built a simulated dataset by integrating the Soil Canopy Observation, Photochemistry and Energy fluxes v1.7 (SCOPE v1.7) [28] and the MODerate resolution atmospheric TRANsmission 5 (MODTRAN 5) [29] models. The SCOPE model was used for the simulation of the canopy reflectance and SIF spectra, while the MODTRAN 5 model was used for the simulation of the atmospheric radiation transfer parameters. Four different levels of chlorophyll content and leaf area index (LAI), and five different typical leaf inclination distribution conditions were used in the SCOPE model to represent different (80 different simulated samples in total) vegetation conditions. For the O2-A band we studied in this paper, the absorption by oxygen is the dominant characteristics of atmospheric radiation transfer, and is mainly influenced by the observation geometry. So the atmospheric condition (including aerosol model, aerosol optical depth, water vapor and ozone column, etc.) were set as fixed values. The main parameters used in the SCOPE and MODTRAN 5 models are listed in Table 1. All other parameters were set to their default values. The spectral resolution (full width at half maximum) of the simulated spectra was 0.3 nm and the spectral sampling interval was 0.15 nm. Therefore, the modeling of the footprint and ERTPL of bi-hemispherical spectral measurements is important in the context of tower-based measurements of SIF. The aims of this paper are: (1) to analyze the footprints of tower-based hemispherical spectral observations, and also the match between these footprints and those of tower-based flux observations; (2) to determine the ERTPL of hemispherical spectral observations and; (3) to evaluate the derived ERTPL using simulations and to test the validity of the atmospheric correction by using the derived ERTPL for tower-based SIF retrieval.

Data Simulation
In order to model the ERTPL, we built a simulated dataset by integrating the Soil Canopy Observation, Photochemistry and Energy fluxes v1.7 (SCOPE v1.7) [28] and the MODerate resolution atmospheric TRANsmission 5 (MODTRAN 5) [29] models. The SCOPE model was used for the simulation of the canopy reflectance and SIF spectra, while the MODTRAN 5 model was used for the simulation of the atmospheric radiation transfer parameters. Four different levels of chlorophyll content and leaf area index (LAI), and five different typical leaf inclination distribution conditions were used in the SCOPE model to represent different (80 different simulated samples in total) vegetation conditions. For the O 2 -A band we studied in this paper, the absorption by oxygen is the dominant characteristics of atmospheric radiation transfer, and is mainly influenced by the observation geometry. So the atmospheric condition (including aerosol model, aerosol optical depth, water vapor and ozone column, etc.) were set as fixed values. The main parameters used in the SCOPE and MODTRAN 5 models are listed in Table 1. All other parameters were set to their default values. The spectral resolution (full width at half maximum) of the simulated spectra was 0.3 nm and the spectral sampling interval was 0.15 nm. Assuming the surface is Lambertian, with the simulation results from SCOPE and MODTRAN 5, the upwelling radiance at the height of the observation platform can be calculated as [30]: where L H is the upwelling radiance arriving at the observation platform at a height H above the target surface; L 0 is the radiance at the top of the canopy, which contains a contribution from both reflected radiance and emitted SIF; T ↑ is the upwelling transmittance of the atmosphere; ρ is the surface reflectance; S is the atmospheric spherical albedo accounting for multiple scattering between atmosphere and surface; and L path is the atmospheric path radiance. For tower-based observations, the platform height is relatively low. The atmospheric scattering effect was, therefore, neglected in this study and Equation (1) can be simplified as: Using Equation (2), L H for different values of the VZA and RAA can be calculated using the simulations made by SCOPE and MODTRAN 5.

Footprint of Spectral Observation
The measured upwelling irradiance (E) can be calculated by integrating the radiance (L) over all the propagation directions (defined using the view zenith angle (θ) and view azimuth angle (ϕ)) within the FOV. For hemispherical measurements, neglecting the influence of atmospheric scattering, the measured irradiance of the signal within the FOV can be expressed as [13,31,32]: where T ↑ (θ) is the upwelling atmospheric transmittance at the given view zenith angle, θ. For bands free from atmospheric absorption, the transmittance can be assumed independent of the RTPL (i.e., T ↑ is independent of θ). Assuming that the ground surface is homogeneous and Lambertian (i.e., L is independent of the propagation direction), Equation (3) can be expressed as: According to Equation (4), for hemispherical measurements made with the cosine-corrected foreoptic, the measured irradiance of the signal within the FOV is proportional to the square of sin θ. Using Equation (4) and discretizing the FOV to 1 • intervals, it is possible to model the fractional contribution of the signal originating from each 1 • interval of the zenith angle to the hemispherical irradiance (see the blue circles in Figure 2), as well as the accumulated contribution of the signal within the FOV (see the red diamonds in Figure 2). It can be seen from Figure 1 that the signal within an FOV of 72 • contributes 90% of the total hemispherical irradiance, of which the maximum contribution comes from a view zenith angle of around 45 • . According to Equation (4), for hemispherical measurements made with the cosine-corrected foreoptic, the measured irradiance of the signal within the FOV is proportional to the square of sin . Using Equation (4) and discretizing the FOV to 1° intervals, it is possible to model the fractional contribution of the signal originating from each 1° interval of the zenith angle to the hemispherical irradiance (see the blue circles in Figure 2), as well as the accumulated contribution of the signal within the FOV (see the red diamonds in Figure 2). It can be seen from Figure 1 that the signal within an FOV of 72° contributes 90% of the total hemispherical irradiance, of which the maximum contribution comes from a view zenith angle of around 45°.

ERTPL Modeling Based on Theoretical Derivation
The RTPL for hemispherical observations made with cosine-corrected foreoptic is more complex than for conical observations made using a bare fiber. When the FOV is as large as 180°, the ERTPL can no longer be estimated as being equal to the height of the sensor above the target surface. For spectral measurements made at the atmospheric windows, the influence of atmospheric radiation transfer on tower-based observations can be ignored as the height of the sensor above the target surface is usually only tens of meters. However, SIF observations are made at the atmospheric absorption bands and these are very sensitive to the RTPL. Therefore, for hemispherical observations, accurate modeling of the ERTPL is important.
According to Equation (3), the upwelling hemispherical irradiance observed with the cosinecorrected foreoptic can be expressed as: Assuming that the surface is homogeneous and isotropic, and ignoring the bidirectional reflectance effect of the surface and the directional characteristics of SIF emission (i.e., is independent of and ), the observed upwelling irradiance can be expressed as: On the other hand, the observed upwelling irradiance can be expressed as the product of the upwelling irradiance at top of canopy ( ) and the equivalent transmittance between the canopy and the sensor ( ↑ ):

ERTPL Modeling Based on Theoretical Derivation
The RTPL for hemispherical observations made with cosine-corrected foreoptic is more complex than for conical observations made using a bare fiber. When the FOV is as large as 180 • , the ERTPL can no longer be estimated as being equal to the height of the sensor above the target surface. For spectral measurements made at the atmospheric windows, the influence of atmospheric radiation transfer on tower-based observations can be ignored as the height of the sensor above the target surface is usually only tens of meters. However, SIF observations are made at the atmospheric absorption bands and these are very sensitive to the RTPL. Therefore, for hemispherical observations, accurate modeling of the ERTPL is important.
According to Equation (3), the upwelling hemispherical irradiance observed with the cosine-corrected foreoptic can be expressed as: Assuming that the surface is homogeneous and isotropic, and ignoring the bidirectional reflectance effect of the surface and the directional characteristics of SIF emission (i.e., L 0 is independent of θ and ϕ), the observed upwelling irradiance can be expressed as: On the other hand, the observed upwelling irradiance can be expressed as the product of the upwelling irradiance at top of canopy (E 0 ) and the equivalent transmittance between the canopy and the sensor (T ↑ ): Combining Equations (6) and (7), the equivalent transmittance can be calculated as: For the oxygen absorption bands, the transmittance is mainly related to the RTPL. Figure 3 shows the variation of the transmittance of the up-welling radiation with RTPL, based on simulations made by MODTRAN 5 with different aerosol optical depth (AOD 550 is set as 0.1, 0.3, and 0.5). As there is no absorption by water and ozone at the wavelength of 761.1 nm we studied, the water vapor column is fixed as 3 g/cm 2 , and the ozone column is fixed as 300 DU. The view zenith angle is set to the range 0-72 • with a sensor height of 20 m above the target surface, and the corresponding range of the RTPL is about 20-65 m. The results show that the transmittance is approximately linearly related to the RTPL within the range we tested (with R 2 > 0.99). As shown in Figure 2, the signal within an FOV of 72 • contributes more than 90% of the total hemispherical irradiance. So when estimating the equivalent transmittance for hemispherical observations, it is reasonable to assume a linear relation between T ↑ and RTPL when the atmospheric condition is fixed. Based on this assumption, T ↑ and T ↑ (θ) in Equation (8) can be replaced by the ERTPL and RTPL (θ): where H is the height of the sensor above the target surface. According to this analysis, we can conclude that the ERTPL of hemispherical spectral observations with cosine-corrected foreoptic is about twice the height of the sensor above the target surface.
Combining Equations (6) and (7), the equivalent transmittance can be calculated as: For the oxygen absorption bands, the transmittance is mainly related to the RTPL. Figure 3 shows the variation of the transmittance of the up-welling radiation with RTPL, based on simulations made by MODTRAN 5 with different aerosol optical depth (AOD550 is set as 0.1, 0.3, and 0.5). As there is no absorption by water and ozone at the wavelength of 761.1 nm we studied, the water vapor column is fixed as 3 g/cm 2 , and the ozone column is fixed as 300 DU. The view zenith angle is set to the range 0-72° with a sensor height of 20 m above the target surface, and the corresponding range of the RTPL is about 20-65 m. The results show that the transmittance is approximately linearly related to the RTPL within the range we tested (with R 2 > 0.99). As shown in Figure 2, the signal within an FOV of 72° contributes more than 90% of the total hemispherical irradiance. So when estimating the equivalent transmittance for hemispherical observations, it is reasonable to assume a linear relation between ↑ and RTPL when the atmospheric condition is fixed. Based on this assumption, ↑ and ↑ ( ) in Equation (8) can be replaced by the ERTPL and RTPL ( ): where is the height of the sensor above the target surface. According to this analysis, we can conclude that the ERTPL of hemispherical spectral observations with cosine-corrected foreoptic is about twice the height of the sensor above the target surface.    Table 1.

Matching between the Footprints of Spectral and Flux Observations
To investigate the matching between the footprints of spectral and flux observations, the footprint of the flux measurements needs to be modeled. In contrast to spectral observations, estimating the footprint of flux measurements is more complex due to the influence of the atmosphere. Kljun et al. [25] presented a two-dimensional parameterisation for flux footprint prediction (FFP). Unlike other existing fast analytical footprint models, the FFP parameterisation is valid for a wide range of boundary layer stratifications and receptor heights, as well as for non-Gaussian turbulence. Therefore, in this study, FFP was employed for the simulation of the flux measurement footprint. The Xiao Tangshan EC site, located at the National Precision Agriculture Demonstration Base in the town of Xiao Tangshan, north of Beijing, China (40.17 • N,116.39 • E), was selected as a test site, and parameters measured on 18 April 2016 were used (details listed in Table 2). Using the online FFP (http://footprint.kljun.net/), a two-dimensional discrete footprint function at a height of 20 m for convective conditions was then modeled-shown as the red lines in Figure 3. According to Equation (4), it is possible to model the cumulative footprint contours of the hemispherical spectral observations made at nadir at a height of 20 m above the target surface. These are shown as the black lines in Figure 3. On the other hand, for the conical observations, the radiance measured by the bare fiber at nadir comes from a circular area on the ground with a radius of where H is the height of the sensor above the target surface. The source area for these measurements is marked as the blue circular area in Figure 3.
As Figure 4 shows, for a typical sensor height above the target surface of 20 m, 90% of the total radiation comes from an FOV lying within 72 • (the corresponding footprint radius is 61.55 m), which can cover 75.68% of the source area of the total flux signal under convective conditions. In contrast, the total surface source area of the conical observations (FOV of 25 • ) is a circle with a radius of 4.43 m, which covers only 1.93% of the flux footprint. Therefore, compared to the conical observation, the hemispherical observation has its advantages in the coordinated measurements of spectra and flux.

Matching between the Footprints of Spectral and Flux Observations
To investigate the matching between the footprints of spectral and flux observations, the footprint of the flux measurements needs to be modeled. In contrast to spectral observations, estimating the footprint of flux measurements is more complex due to the influence of the atmosphere.
Kljun et al. [25] presented a two-dimensional parameterisation for flux footprint prediction (FFP). Unlike other existing fast analytical footprint models, the FFP parameterisation is valid for a wide range of boundary layer stratifications and receptor heights, as well as for non-Gaussian turbulence. Therefore, in this study, FFP was employed for the simulation of the flux measurement footprint. The Xiao Tangshan EC site, located at the National Precision Agriculture Demonstration Base in the town of Xiao Tangshan, north of Beijing, China (40.17° N,116.39° E), was selected as a test site, and parameters measured on 18 April 2016 were used (details listed in Table 2). Using the online FFP (http://footprint.kljun.net/), a two-dimensional discrete footprint function at a height of 20 m for convective conditions was then modeled-shown as the red lines in Figure 3. According to Equation (4), it is possible to model the cumulative footprint contours of the hemispherical spectral observations made at nadir at a height of 20 m above the target surface. These are shown as the black lines in Figure 3. On the other hand, for the conical observations, the radiance measured by the bare fiber at nadir comes from a circular area on the ground with a radius of • tan( ), where is the height of the sensor above the target surface. The source area for these measurements is marked as the blue circular area in Figure 3.

Evauation of the Modeled ERTPL Using Simulations
Using the simulated irradiance and transmittance at different VZAs, the modeled ERTPL can be evaluated. It should be noted that the ERTPL model is based on the assumption that the surface is isotropic and that the transmittance is linearly related to the RTPL. However, in practice, these assumptions are not valid. First, for typical vegetation, both the reflectance and the SIF emission varies with the viewing and illumination directions [33]. Secondly, the relationship between the transmittance and RTPL can only be modeled by linear functions for a limited range of the view zenith angle. When the view zenith angle is close to 90 • , clearly this relation will become nonlinear (as Figure 4 shows). Therefore, in practice, the ERTPL will not be exactly as shown in Equation (9). Figure 5 shows the variation in the simulated up-welling radiance at top of canopy and atmospheric transmittance of typical vegetation (with an LAI of 4, chlorophyll content of 40 µg/cm 2 , and LIDF of spherical) inside the O 2 -A absorption band (761.1 nm) at different view zenith angles across the solar principal plane simulated by the SCOPE and MODTRAN 5 models. The directional characteristics of up-welling radiance at top of canopy is caused by the bidirectional reflectance effect of the canopy and the directional emission of SIF. The bidirectional reflectance effect of canopy has been widely studied, and has an obvious bowl-edge effect at the far-red band [33,34]. The emission of SIF has also been proved to have similar directional distribution characteristics as reflectance for both observations and simulations [33]. This means that the up-welling radiance at top of canopy will increase as the view zenith angle increases, as shown in Figure 5. In contrast, at the oxygen absorption band, the atmospheric transmittance will fall as the view zenith angle (or RTPL) increases. In other words, the directional characteristics of the up-welling radiance and atmospheric transmittance have opposing influences on the observed upwelling radiance. Therefore, the errors caused by the two assumptions that were made in modeling the ERTPL of the hemispherical measurements will not offset each other to some extent (at least will not accumulate), and the accuracy of the modeled ERTPL value of 2H will be reasonable. radiation comes from an FOV lying within 72° (the corresponding footprint radius is 61.55 m), which can cover 75.68% of the source area of the total flux signal under convective conditions. In contrast, the total surface source area of the conical observations (FOV of 25°) is a circle with a radius of 4.43 m, which covers only 1.93% of the flux footprint. Therefore, compared to the conical observation, the hemispherical observation has its advantages in the coordinated measurements of spectra and flux.

Evauation of the Modeled ERTPL Using Simulations
Using the simulated irradiance and transmittance at different VZAs, the modeled ERTPL can be evaluated. It should be noted that the ERTPL model is based on the assumption that the surface is isotropic and that the transmittance is linearly related to the RTPL. However, in practice, these assumptions are not valid. First, for typical vegetation, both the reflectance and the SIF emission varies with the viewing and illumination directions [33]. Secondly, the relationship between the transmittance and RTPL can only be modeled by linear functions for a limited range of the view zenith angle. When the view zenith angle is close to 90°, clearly this relation will become nonlinear (as Figure 4 shows). Therefore, in practice, the ERTPL will not be exactly as shown in Equation (9). Figure 5 shows the variation in the simulated up-welling radiance at top of canopy and atmospheric transmittance of typical vegetation (with an LAI of 4, chlorophyll content of 40 μg/cm 2 , and LIDF of spherical) inside the O2-A absorption band (761.1 nm) at different view zenith angles across the solar principal plane simulated by the SCOPE and MODTRAN 5 models. The directional characteristics of up-welling radiance at top of canopy is caused by the bidirectional reflectance effect of the canopy and the directional emission of SIF. The bidirectional reflectance effect of canopy has been widely studied, and has an obvious bowl-edge effect at the far-red band [33,34]. The emission of SIF has also been proved to have similar directional distribution characteristics as reflectance for both observations and simulations [33]. This means that the up-welling radiance at top of canopy will increase as the view zenith angle increases, as shown in Figure 5. In contrast, at the oxygen absorption band, the atmospheric transmittance will fall as the view zenith angle (or RTPL) increases. In other words, the directional characteristics of the up-welling radiance and atmospheric transmittance have opposing influences on the observed upwelling radiance. Therefore, the errors caused by the two assumptions that were made in modeling the ERTPL of the hemispherical measurements will not offset each other to some extent (at least will not accumulate), and the accuracy of the modeled ERTPL value of 2H will be reasonable.  Using the simulated values of L H and L 0 for different VZA and RAA, the irradiance observed by a hemispherical measurement system with cosine-corrected foreoptic at a height H (E cos ) above the canopy and at the top of canopy (E 0 ) can be calculated by integration using Equations (5) and (7). Hence, the equivalent atmospheric transmittance can be calculated as: For bands at atmospheric windows, the influence of atmospheric absorption is very weak. The purpose of this study was to analyze the effect of the atmospheric radiation transfer at the oxygen absorption bands on the SIF retrieval. Therefore, the O 2 -A band (centered at 761.1 nm in the simulated dataset), which is frequently used in SIF retrieval, was selected to evaluate the modeled ERTPL. According to Equation (10), the equivalent atmospheric transmittance of the spectral observations with cosine-corrected foreoptic at a height above the canopy of 20 m is 0.924, and the corresponding ERTPL is 37.7 m (~1.89H), which is close to the modeled ERTPL of 2H.
The accuracy of atmospheric correction of the hemispherical observation of irradiance at 761.1 nm (within the O 2 -A absorption band) at a height of H above the canopy with the ERTPL of 1.89H or 2H was evaluated by comparing with the simulated reference irradiance at top of canopy, as shown in Figure 6. The scatters of both corrected irradiance with ERTPL of 1.89H and 2H locate close to the 1:1 line, and the RRMSEs are 0.16% and 0.55%, respectively. The difference between the performance of ERTPL of 1.89H and 2H is quite tiny. The results indicate that the ERTPL of 2H modeled in this study is efficient for the atmospheric correction for tower-based hemispherical observation of up-welling irradiance. Using the simulated values of and for different VZA and RAA, the irradiance observed by a hemispherical measurement system with cosine-corrected foreoptic at a height H ( ) above the canopy and at the top of canopy ( ) can be calculated by integration using Equations (5) and (7). Hence, the equivalent atmospheric transmittance can be calculated as: For bands at atmospheric windows, the influence of atmospheric absorption is very weak. The purpose of this study was to analyze the effect of the atmospheric radiation transfer at the oxygen absorption bands on the SIF retrieval. Therefore, the O2-A band (centered at 761.1 nm in the simulated dataset), which is frequently used in SIF retrieval, was selected to evaluate the modeled ERTPL. According to Equation (10), the equivalent atmospheric transmittance of the spectral observations with cosine-corrected foreoptic at a height above the canopy of 20 m is 0.924, and the corresponding ERTPL is 37.7 m (~1.89H), which is close to the modeled ERTPL of 2H.
The accuracy of atmospheric correction of the hemispherical observation of irradiance at 761.1 nm (within the O2-A absorption band) at a height of H above the canopy with the ERTPL of 1.89H or 2H was evaluated by comparing with the simulated reference irradiance at top of canopy, as shown in Figure 6. The scatters of both corrected irradiance with ERTPL of 1.89H and 2H locate close to the 1:1 line, and the RRMSEs are 0.16% and 0.55%, respectively. The difference between the performance of ERTPL of 1.89H and 2H is quite tiny. The results indicate that the ERTPL of 2H modeled in this study is efficient for the atmospheric correction for tower-based hemispherical observation of upwelling irradiance.

Performance of the Atmospheric Correction Using the Modeled ERTPL for SIF Retrieval from the Simulated Dataset
As the fundamental objective of the analysis of the ERTPL of the hemispherical spectral observations was the retrieval of the SIF from tower-based observations, the retrieved SIF from the simulated spectral dataset with 80 different vegetation conditions (as described in Section 2.1) were employed to evaluate the accuracy of the modelled ERTPL.
The Fraunhofer Line Discrimination (FLD) principle [35] is the mainly used methodology for SIF retrieval at canopy level. Several different FLD-based algorithms have been proposed and applied, such as the standard FLD [35], the 3-bands FLD (3FLD) [36], the improved FLD (iFLD) [37], the

Performance of the Atmospheric Correction Using the Modeled ERTPL for SIF Retrieval from the Simulated Dataset
As the fundamental objective of the analysis of the ERTPL of the hemispherical spectral observations was the retrieval of the SIF from tower-based observations, the retrieved SIF from the simulated spectral dataset with 80 different vegetation conditions (as described in Section 2.1) were employed to evaluate the accuracy of the modelled ERTPL.
The Fraunhofer Line Discrimination (FLD) principle [35] is the mainly used methodology for SIF retrieval at canopy level. Several different FLD-based algorithms have been proposed and applied, such as the standard FLD [35], the 3-bands FLD (3FLD) [36], the improved FLD (iFLD) [37], the principal components analysis based FLD (pFLD) [38], etc. Besides, some spectral fitting methods (SFM) were also proposed and have been proved to be more reliable for SIF retrieval from spectral data with relatively low spectral resolution and signal-to-noise ratio. According to the study by Liu et al. [39], the 3FLD algorithm is robust and simple for SIF retrieval from data with spectral resolution of 0.3 nm, and only three spectral samples are needed. So the 3FLD method was selected for the SIF retrieval from the simulated dataset (with no noise) in this study. Using the 3FLD method, the SIF can be calculated as: where w is the weight of the band and is inversely proportion to the distance between the left-hand/right-hand band and the inner band; I is the downwelling irradiance arriving at the TOC; L is the total upwelling radiance at the TOC; and the subscripts "in", "left" and "right" refer to the bands inside, at the left of and at the right of the absorption band, respectively. SIF retrieved using radiation simulations at the top of the canopy (SIF TOC ), tower-based SIF retrieved using original spectral simulations at a height of H (20 m) without atmospheric correction (SIF H ), and tower-based SIF retrieved using atmospherically corrected spectra and RTPL values of H and 2H (SIF corr_H and SIF corr_2H ) were compared with the reference values of the simulated SIF, as shown in Figure 7. principal components analysis based FLD (pFLD) [38], etc. Besides, some spectral fitting methods (SFM) were also proposed and have been proved to be more reliable for SIF retrieval from spectral data with relatively low spectral resolution and signal-to-noise ratio. According to the study by Liu et al. [39], the 3FLD algorithm is robust and simple for SIF retrieval from data with spectral resolution of 0.3 nm, and only three spectral samples are needed. So the 3FLD method was selected for the SIF retrieval from the simulated dataset (with no noise) in this study. Using the 3FLD method, the SIF can be calculated as: where w is the weight of the band and is inversely proportion to the distance between the lefthand/right-hand band and the inner band; I is the downwelling irradiance arriving at the TOC; L is the total upwelling radiance at the TOC; and the subscripts "in", "left" and "right" refer to the bands inside, at the left of and at the right of the absorption band, respectively. SIF retrieved using radiation simulations at the top of the canopy (SIFTOC), tower-based SIF retrieved using original spectral simulations at a height of H (20 m) without atmospheric correction (SIFH), and tower-based SIF retrieved using atmospherically corrected spectra and RTPL values of H and 2H (SIFcorr_H and SIFcorr_2H) were compared with the reference values of the simulated SIF, as shown in Figure 7.  RRMSE=293.79% Figure 7. The relation between the modelled reference SIF and the modelled SIF retrieved by the 3FLD method using different atmospheric correction methods. SIF TOC is the SIF retrieved using radiation simulations at the top of the canopy; SIF H is the tower-based SIF retrieved using original spectral simulations at a height of H (20 m) without atmospheric correction; and SIF corr_H and SIF corr_2H are the tower-based SIF retrieved using atmospherically corrected spectra and RTPL values of H and 2H, respectively.
As shown in Figure 7, all the SIF H values are negative and located far from the 1:1 line (with a relative root mean square error (RRMSE) of 293.79%), which means that the atmospheric correction was necessary for the tower-based SIF observations although the height of sensor was only 20 m. Compared to the SIF H values, the SIF corr_H values are located much closer to the 1:1 line but the errors are still high (RRMSE = 133.71%). In contrast, the SIF corr_2H values are located close to the 1:1 line (RRMSE = 18.22%) as are the SIF TOC values (RRMSE = 17.47%). It needs to be noted that the ERTPL model proposed in this study overestimates the ERTPL to some extent and, consequently, leads to some overestimation of the SIF. These results indicate that an ERTPL of 2H is suitable for the atmospheric correction of hemispherical observations.

Discussion
The relationship between SIF and GPP is still not very clear, and there are a lot of uncertainties in both the mechanism and the observations [6]. In this paper, we focused on the match between the footprints of SIF and flux observations. For the tower-based observation of up-welling irradiance of vegetation, there are mainly two different configurations: conical observation and hemispherical observation [8,26]. We compared the footprints of the spectral observations of the two configurations, and modeled the ERTPL of hemispherical observations for atmospheric correction at the oxygen absorption band.
In recent years, more and more automatic tower-based spectral observation systems were established to obtain long-term observations for vegetation in coordination with flux measurements for linking remotely sensed data to ecosystem characteristics [12][13][14][15][16][17]. For those observations, one of the scientific challenges is to determine the most suitable FOV of spectral observations to match with the flux footprint. Balzarolo et al. [11] reviewed the configuration of 55 optical systems at 42 flux tower sites in 2011, and found that 17 out of the 55 systems used hemispherical observations of up-welling irradiance, and the others used the conical observations with FOVs from 5 • to 60 • .
Porcar-Castell et al. [10] claimed that the hemispherical measurements had great advantage of enabling the sampling of a wider area. According to the results of this paper, 90% of the total radiation comes from an FOV of width 72 • , which in turn covered 75.68% of the source area of the flux measurements. So the hemispherical measurements have an obvious advantage to match with the flux footprint. For conical measurements, some alternative ways for better matching with the flux footprint were also proposed. For example, Hilker et al. [40,41] used spectral observations collected over a circular area centered at the flux tower with a rotating system; Gamon et al. [42] introduced a tram system to make spatially representative observations within the flux footprint.
However, there are also a lot of disadvantages for the hemispherical spectral measurements. Firstly, the tower body and its shadow will be in the field of view. Moreover, the influence of them will vary with the illumination geometry [10], which will cause some uncertainties in the observation. For example, if the diameter of the projection of the tower body on the ground is 5 m, and the height of sensor is 20 m, the tower body will cover a FOV of about 14.25 • (about 8% of the total FOV of the hemispherical observation). Secondly, the surface of vegetation canopy is not isotropic. Both the reflectance and SIF are directional [33]. For hemispherical observations, radiance from all directions with different weight will be collected. So the directional characteristics of reflectance or SIF should be carefully considered for different cases of application. Thirdly, as the transmittance of cosine-corrected foreoptic is limited (about 25-30% at the far-red band for the Ocean Optics CC-3 cosine corrector), the signal-to-noise ratio of the hemispherical spectral observations will decrease to some extent, which, clearly, could reduce the accuracy of the SIF retrieval.
Another important issue to be considered for the hemispherical observation is the more complex path of radiation transfer. For the conical observation, the FOV is usually very small, and the RTPL can be estimated as the height of the sensor. But for the hemispherical observation, the FOV is 180 • , and the ERTPL would be much longer than the sensor's height. For the tower-based observations for reflectance, wavelengths at atmospheric windows are usually used, so the influence of atmospheric radiation transfer is usually neglected. However, for SIF observation, the atmospheric absorption bands are needed, and the influence of atmosphere is significant [27]. So the ERTPL for tower-based hemispherical observations needs to be modeled. In this paper, according to the theoretical derivation and evaluation with simulated dataset, the ERTPL is modeled as twice of the sensor's height. The retrieved SIF values from the corrected irradiance using the modeled ERTPL fit well with the simulated reference SIF values.
It needs to be noted that, the result of the ERTPL for hemispherical observation acquired in this study relies on some assumptions. Firstly, the surface reflectance and SIF emission is assumed to be isotropic whereas, in practice, both reflectance and SIF emission of vegetation has obvious directional characteristics. Moreover, the surface is usually heterogeneous. Secondly, the atmospheric transmittance is assumed to be linearly related to the RTPL, which is only true for a limited range of the view zenith angle. Thirdly, for common setups of tower-based observation, the height of sensor is only tens of meters, and the atmospheric scattering effect is tiny. So the effects of atmospheric scattering were neglected. According to our analysis, the errors caused by the former two assumptions are opposite at the far-red band. Although they cannot totally offset each other, the error will not be accumulated. Finally, this study is totally based on simulation due to the lack of field measurements. Further analysis based on measured dataset should be carried out.

Conclusions
Using tower-based spectral measurements in coordination with flux measurements is an efficient way of linking SIF to the photosynthesis status. For the observation of up-welling irradiance, both the conical and the hemispherical configurations have their own advantages, but the footprint of hemispherical observation is much wider than the conical observation, and would surely better match with the footprint of flux measurement. However, the effect of atmospheric radiation transfer for hemispherical observation is more complex. In this paper, we developed and evaluated the models of the footprint and ERTPL of hemispherical spectral observations using a simulated dataset and evaluated the performance of the atmospheric correction by using the modeled ERTPL for simulated tower-based SIF retrieval.
First, we developed a method of modeling the footprint of hemispherical spectral observations and found that 90% of the total radiation comes from an FOV of width 72 • (the corresponding radius of the footprint is about 3.1 times the sensor's height above the target surface). For a typical instrument installation height of 20 m above the target surface, and given convective conditions with light winds (0.74 m/s), 90% of the radiation contributing to the hemispherical spectral observations originates from an area that covers 75.68% of the source area of the flux measurements. For conical spectral observations, in contrast, the footprint covers just 1.93% of the flux source area. These results indicate that, when made in conjunction with flux measurements, hemispherical spectral measurements are superior to conical measurements.
Second, we built a model to estimate the ERTPL of hemispherical spectral observations. Assuming the surface is isotropic and the transmittance is linearly related to the RTPL, the ERTPL of hemispherical spectral observations with cosine-corrected foreoptic can be estimated as being equal to twice of the sensor's height above the target surface. The modeled ERTPL was evaluated using simulations made by SCOPE and MODTRAN 5. Taking the directional characteristics of up-welling radiance at top of canopy and the non-linear relationship between atmospheric transmittance and RTPL into account, the calculated ERTPL based on the simulations was 1.89H, which is close to the modeled value of 2H. These results indicate that the ERTPL model described in this paper is suitable for making atmospheric corrections to tower-based spectral observations. Furthermore, the SIF retrieval results based on the simulations also indicate that the modeled ERTPL of 2H is acceptable for use in atmospheric correction. The SIF was retrieved by the 3FLD method using simulated spectra at the top of the canopy, and spectra observed at a height of 20 m without atmospheric correction and with atmospheric corrections using an RTPL of H and 2H. The SIF values retrieved using spectra atmospherically corrected using an RTPL of 2H matched the reference SIF values well-the RRMSE was 18.22%. For the SIF retrieved from spectra that were atmospherically corrected using an RTPL of H and from spectra without atmospheric correction, the RRMSEs were 133.71% and 293.79%, respectively, indicating totally unreliable results. Therefore, the ERTPL model proposed in this paper is helpful for SIF retrieval based on hemispherical spectral measurements.
In conclusion, considering the match between the footprints of spectral and flux measurements, the hemispherical configuration for the observation of up-welling irradiance has advantage, and the ERTPL for hemispherical observation can be estimated as twice of the sensor's height above the surface.